Persistence of Anderson Localization in Schroedinger Operators With Decaying Random Potentials

摘要

We show persistence of both Anderson and dynamical localization in Schroedinger operations with non-positive (attractive) random decaying potential. We consider an Anderson-type Schroedinger operator with a non- positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than absolute value of x (exp -2) at infinity, we prove that the operator has infinitely many eigen-values below zero. For envelopes decaying as absolute value of x (exp -alpha) at infinity, we determine the number of bound states below a given energy EPSILON < OMICRON, asymptotically as alpha darr OMICRON. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent alpha; (b) dynamical localization holds uniformly in alpha.